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SIMPLIFIED MODEL for REINFORCED CONCRETE BEAMS UNDER CATENARY ACTION a Thesis Presente SIMPLIFIED MODEL FOR REINFORCED CONCRETE BEAMS UNDER CATENARY ACTION _______________________________________ A Thesis presented to the Faculty of the Graduate School at the University of Missouri-Columbia _______________________________________________________ In Partial Fulfillment of the Requirements for the Degree Master of Science _____________________________________________________ by RASOLOFOMALALA ANDRY NIRINA Dr. Sarah Orton, Thesis Supervisor JULY 2010 The undersigned, appointed by the dean of the Graduate School, have examined the thesis entitled SIMPLIFIED MODEL FOR REINFORCED CONCRETE BEAMS UNDER CATENARY ACTION presented by Rasolofomalala Andry Nirina, a candidate for the degree of Master of Science in Civil Engineering, and hereby certify that, in their opinion, it is worthy of acceptance. Professor Sarah Orton Professor Hani Salim Professor P. Frank Pai ACKNOWLEDGEMENTS I would like to thank the Fulbright Visiting Student Junior Staff Development Program for having given me such a great opportunity to complete my degree program in the U.S. and fulfill my life dream of getting my degree in Structural Engineering. I express my deepest gratitude to my Academic Advisor Assoc. Prof. Sarah Orton for her guidance, advice, criticism, encouragements, and insight throughout the research. I could not have asked for advisors more generous with her time and consideration. My gratitude is extended to the other faculty members, Prof. P. Frank Pai and Prof. Hani Salim for their valuable suggestions and comments and for serving on my thesis committee. I also express my thanks and appreciation to my family for their understanding, support and patience during those two years. Lastly, I am thankful to all professors, staff and friends who have contributed to my education and made my stay at the University of Missouri a memorable and valuable experience. ii ABSTRACT The purpose of this research is to develop simplified models to predict the load displacement behavior for any reinforced concrete beam on the basis of material, geometric, and design parameters. The proposed simplified model includes a beam element and a system of springs that represents the load extension behavior and moment rotation behavior of a reinforced concrete beam element. Spring properties are based on material, geometric, and design parameters. The tensile definition of the axial extension spring is approximated by a steel stress-strain curve modified to account for uneven stress in the steel along the length of the beam. The commercial finite element code ANSYS is used to analyze the models. The results of the models are then compared with experimental tests on reinforced concrete beams, and good agreements are found. iii TABLE OF CONTENTS ACKNOWLEDGEMENTS ............................................................................................................ ii ABSTRACT ................................................................................................................................... iii LIST OF FIGURES ....................................................................................................................... vi LIST OF TABLES .......................................................................................................................... x 1. INTRODUCTION ............................................................................................................... 1 1.1 Problem ......................................................................................................................... 3 1.2 Objectives and research approach ................................................................................ 3 1.3 Outline .......................................................................................................................... 4 2. LITERATURE REVIEW .................................................................................................... 6 2.1 Disproportionate collapse in buildings ......................................................................... 6 2.2 Modeling of reinforced concrete beams under catenary action .................................. 13 2.3 Experimental tests of reinforced concrete beams under catenary action .................... 16 3. MODELLING OF REINFORCED CONCRETE BEAM UNDER CATENARY ACTION 28 3.1 Response of a reinforced concrete beam under catenary action ................................. 28 3.2 Proposed simplified model ......................................................................................... 31 3.3 Definition of the elements of the simplified model .................................................... 34 3.4 Implementation of the simplified models in ANSYS ................................................. 50 4. COMPARISON OF SIMPLIFIED MODEL RESULTS AND PREVIOUS EXPERIMENTAL TEST RESULTS ........................................................................................... 70 4.1 Model with non-continuous reinforcement – Orton beams NR-2 and PM-2 ............. 70 4.2 Model with continuous reinforcement – Orton beams NM-1 and NM-2 ................... 73 iv 4.3 Model with continuous reinforcement – Sasani and Kropelnicki beam ..................... 76 5. CONCLUSIONS................................................................................................................ 78 5.1 Conclusion .................................................................................................................. 78 5.2 Future Research .......................................................................................................... 79 REFERENCES ............................................................................................................................. 81 v LIST OF FIGURES Figure 1-1.Beam under catenary action .......................................................................................... 2 Figure 2-1.Ronan Point building after collapse (Nair, 2005) ......................................................... 7 Figure 2-2.Murrah Federal Office Building after 19 April 1995 attack (Crawford, 2004) ............ 8 Figure 2-3.Pentagon building after the crash damage (Mlakar et al., 2003) ................................ 10 Figure 2-4. Front View of Reinforced Concrete Building Damaged by System Boiler Explosion (Sucuoglu et al., 1994). ................................................................................................................. 11 Figure 2-5. Aerial view of Hotel San Diego ................................................................................. 12 Figure 2-6. South-east view of Hotel San Diego ......................................................................... 12 Figure 2-7. Free body diagram of Regan’s Model (Regan, 1975) ................................................ 13 Figure 2-8. Free body diagram of Orton’s Model (Orton, 2007).................................................. 15 Figure 2-9. Arrangement for PCL catenary test (Regan, 1975). ................................................... 17 Figure 2-10.Results of catenary tests of precast floor strips (Regan, 1975) ................................. 18 Figure 2-11. Reinforcement detailing of the experimental beam (Sasani and Kropelnicki, 2007). ....................................................................................................................................................... 19 Figure 2-12. Force – Displacement relationship (Sasani and Kropelnicki, 2007). ....................... 20 Figure 2-13. Reinforcement design of test specimen (Orton, 2007) ............................................. 21 Figure 2-14. Transfer of catenary forces through stirrups (Orton, 2007) ..................................... 22 Figure 2-15. Vertical and axial loads versus displacement for beams without continuous reinforcement (Orton, 2007). ........................................................................................................ 23 Figure 2-16. Photo of test NR-2 under catenary action ................................................................ 23 Figure 2-17. Vertical and axial loads versus displacement for beams with continuity provided by CFRP sheets (Orton, 2007). .......................................................................................................... 24 vi Figure 2-18. Photo of test PM-2 under catenary action ................................................................ 24 Figure 2-19. The tested four-bay and three-story one-third scale model representing a segment of a larger planar frame structure (Kunnanth, 2008). ................................................................... 26 Figure 2-20. Middle column load versus unloading displacement of failed middle column. (Yi and Kunnanth, 2008) ..................................................................................................................... 27 Figure 3-1. Rotation of rigid block (Orton, 2007) ........................................................................ 29 Figure 3-2 . Transfer of catenary action forces through stirrups (Orton, 2007) ........................... 29 Figure 3-3. Catenary forces provided through the positive and negative moment reinforcement (Orton, 2007)................................................................................................................................. 30 Figure 3-4. a. Reinforced concrete beam with non-continuous reinforcement Orton 2007). ....... 33 Figure 3-5. a. Reinforced concrete beam with continuous reinforcement (Orton 2007). ............. 34
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